Long-time asymptotics for the integrable nonlocal nonlinear Schr\"odinger equation with step-like initial data

Abstract

We study the Cauchy problem for the integrable nonlocal nonlinear Schr\"odinger (NNLS) equation \[ iq_{t}(x,t)+q_{xx}(x,t)+2 q^{2}(x,t)\bar{q}(-x,t)=0 \] with a step-like initial data: $q(x,0)=q_0(x)$, where $q_0(x)=o(1)$ as $x\to-\infty$ and $q_0(x)=A+o(1)$ as $x\to\infty$, with an arbitrary positive constant $A>0$. The main aim is to study the long-time behavior of the solution of this problem. We show that the asymptotics has qualitatively different form in the quarter-planes of the half-plane $-\infty<x<\infty$, $t>0$: (i) for $x<0$, the solution approaches a slowly decaying, modulated wave of the Zakharov-Manakov type; (ii) for $x>0$, the solution approaches the "modulated constant". The main tool is the representation of the solution of the Cauchy problem in terms of the solution of an associated matrix Riemann-Hilbert (RH) problem and the consequent asymptotic analysis of this RH problem.